= Magnetic onset on a rotating velocity branch
{title2=$(\mu_1-es^2)[\mu_2^2+(\omega_2-\omega_3)^2]=2c_2\mu_2s^2,\quad s^2=\mu_3/d_4$}
For real $c_2,d_4$ and the simplified magnetic amplitude equations, the pure-velocity branch is $V=se^{i\omega_3t}$ with $A=B=0$. In its rotating frame the magnetic linearization on $(B,\operatorname{Re}A,\operatorname{Im}A)$ has rows $(\mu_1-es^2,2s,0)$, $(c_2s,\mu_2,-\Delta)$ and $(0,\Delta,\mu_2)$, where $\Delta=\omega_2-\omega_3$. Its zero determinant gives the displayed onset relation. A nonzero magnetic branch needs a kernel involving both magnetic amplitudes, a transverse parameter crossing and nondegenerate nonlinear saturation; determinant zero alone is insufficient in degenerate cases.
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